Topics on Integrable Functions of Real Variables

Abstract

Let $\mu$ be the Lebesgue measure in $\mathbb {R}^N$ and refer the notions of measurability and integrability to such a measure.

Appendices

Problems and Complements

11c Rearranging the Values of a Function

Let f be a real-valued, nonnegative, measurable function, satisfying (11.4).

11.1 :

Prove that the definition of $f^*$ introduced in (11.2) or (11.5) is equivalent to (compare with (15.3) of Chap. 4)

$$\begin{aligned} f^*=\int _0^\infty \chi _{[f>t]}^*dt. \end{aligned}$$

(11.1c)

11.2 :

Give a detailed proof of (iii) of Proposition 11.1.

11.3 :

Prove the following more general version of (vii) of Proposition 11.1. Let $\varphi (\cdot ):\mathbb {R}^+\rightarrow \mathbb {R}^+$ be monotone increasing. Then

$$\begin{aligned} \int _{\mathbb {R}^N}\varphi (f)d\mu =\int _{\mathbb {R}^N}\varphi (f^*)d\mu .\qquad {\mathbf{(vii)}^{\varvec{\prime }}} \end{aligned}$$

11.4 :

Prove that (vii) ${}^{\varvec{\prime }}$ continues to hold for $\varphi =\varphi _1-\varphi _2$, where each of the $\varphi _j$ are monotone increasing and for at least one of them the corresponding integral in (vii) ${}^{\varvec{\prime }}$ is finite.

11.5 :

For a measurable set E of finite measure redefine $E^{*}_c$ as the closed ball about the origin of radius

$$\begin{aligned} \kappa _N R^N=\mu (E). \end{aligned}$$

Then redefine the nondecreasing, symmetric rearrangement $f^{*\prime }_c$ of a nonnegative function f, satisfying (11.4), by the same procedure as in § 11 with the proper modifications. Prove that all statements in Proposition 11.1 continue to hold, except that $f^*_c$ is upper semi-continuous.

11.6 :

If f does not satisfy (11.4) then the symmetric, decreasing rearrangement of f can still be defined by setting

$$\begin{aligned} \chi _{[f>t]}^*=\left\{ \begin{array}{ll} 0\quad &{}\text {if }\> \mu ([f>t])=0;\\ {\displaystyle \chi _{|x|<R}} \quad &{}{\begin{array}{l} \text {if }\>\mu ([f>t])<\infty ;\\ \text {where }\>\kappa _N R^N=\mu ([f>t]); \end{array}}\\ 1\quad &{}\text {if }\quad \mu ([f>t])=\infty . \end{array}\right. \end{aligned}$$

The symmetric rearrangement of f is then defined by the formula (11.1c) using this new definition of $\chi _{[f>t]}^*$. Prove that all statements of Proposition 11.1 remain force.

12c Some Integral Inequalities for Rearrangements

Let f and g be real-valued, nonnegative, measurable functions, satisfying (11.4).

12.1 :

Let E be a measurable set in $\mathbb {R}^N$ of finite measure. Let $B_\rho$ be a ball of radius $\rho$ about the origin and apply (12.1) with $f=\chi _{B_\rho }$ and $g=\chi _E$. Assume that for all balls $B_\rho$ (12.1) holds with equality, i.e.,

$$\begin{aligned} \int _{\mathbb {R}^N} \chi _{B_\rho }\chi _E d\mu = \int _{\mathbb {R}^N} \chi _{B_\rho }^*\chi _E^* d\mu . \end{aligned}$$

Prove that $E=E^*$.

12.2 :

Let $f=f^*$ be strictly decreasing. Prove that (12.1) holds with equality if and only if $g=g^*$.

12.3 :

Let $f=f^*$ be strictly decreasing. Prove that (12.2) holds with equality for all $t\ge 0$ if and only if $g=g^*$.

12.4 :

Prove the following more general version of Theorem 12.1

Theorem 12.1c

Let $\varphi$ be a nonnegative convex function in $\mathbb {R}$ vanishing at the origin. Then

$$\begin{aligned} \int _{\mathbb {R}^N}\varphi (f^*-g^*)d\mu \le \int _{\mathbb {R}^N}\varphi (f-g)d\mu . \end{aligned}$$

(12.1c)

When $\varphi (t)=|t|^p$ for $p\ge 1$ this is Theorem 12.1.

12.5 :

Let $\varphi$ be strictly convex and let $f=f^*$ be strictly decreasing. Prove that (12.1c) holds with equality if and only if $g=g^*$.

20c $L^p$ Estimates of Riesz Potentials

Let E be a domain in $\mathbb {R}^N$ and for $\alpha \in (0,N)$ and $f\in L^p(E)$, consider the potentials

$$\begin{aligned} V_{\alpha ,f}(x)=\int _E\frac{f(y)}{|x-y|^{N-\alpha }}dy. \end{aligned}$$

If $\alpha =1$ these coincide with the Riesz potentials.

Theorem 20.1c

Let $f\in L^p(E)$ for $1<p<\frac{N}{\alpha }$. There exists a constant $C(N,p,\alpha )$ depending only upon N, p and $\alpha$ such that

$$\begin{aligned} \Vert V_{\alpha ,f}\Vert _q\le C(N,p,\alpha ) \Vert f\Vert _p,\qquad \text { where }\quad q= \frac{Np}{N-\alpha p}. \end{aligned}$$

(20.1c)

Compute explicitly the constant $C(N,p,\alpha )$ and verify that it tends to infinity as either $p\rightarrow 1$ or $p\rightarrow \frac{N}{\alpha }$.

21c $L^p$ Estimates of Riesz Potentials for $p=1$ and $p>N$

Proposition 21.1c

For every $\alpha \in (0,N)$ and every $1\le r<\frac{N}{N-\alpha }$

$$\begin{aligned} \sup _{x\in E}\int _E\frac{dy}{|x-y|^{(N-\alpha )r}}\le \frac{\kappa _N^{\frac{N-\alpha }{N}r}}{1-\frac{N-\alpha }{N} r} \mu (E)^{1-\frac{N-\alpha }{N} r}. \end{aligned}$$

(21.1c)

Proposition 21.2c

Let E be of finite measure, and let $f\in L^1(E)$. Then $V_{\alpha ,f}\in L^q(E)$ for all $q\in [1,\frac{N}{N-\alpha })$, and

$$\begin{aligned} \Vert V_{\alpha ,f}\Vert _q\le \frac{\kappa _N^{\frac{N-\alpha }{N}}}{\left( 1-\frac{N-\alpha }{N}q\right) ^{\frac{1}{q}}} \mu (E)^{\frac{1}{q}-\frac{N-\alpha }{N}}\Vert f\Vert _1. \end{aligned}$$

(21.2c)

Remark 21.1c

The limiting integrability $q=\frac{N}{N-\alpha }$ is not permitted in (21.2c).

Proposition 21.3c

Let E be of finite measure, and let $f\in L^p(E)$ for some $p>\frac{N}{\alpha }$. Then $V_{\alpha ,f}\in L^\infty (E)$ and

$$\begin{aligned} \Vert V_{\alpha ,f}\Vert _\infty \le C(N,p,\alpha ) \mu (E)^{\frac{\alpha p-N}{Np}}\Vert f\Vert _p\end{aligned}$$

(21.3c)

where

$$\begin{aligned} C(N,p,\alpha )=\kappa _N^{\frac{N-\alpha }{N}} \Big (\frac{N(p-1)}{\alpha p-N}\Big )^{\frac{p-1}{p}}. \end{aligned}$$

(21.4c)

22c The Limiting Case $p=\frac{N}{\alpha }$

The value $\alpha p=N$ is not permitted neither in Theorem 20.1c nor in Proposition 21.3c. Prove the following:

Theorem 22.1c

Let E be of finite measure, and let $f\in L^p(E)$ for $p=\frac{N}{\alpha }$. There exist constants $C_1$ and $C_2$ depending only upon N and $\alpha$, such that

$$\begin{aligned} \int _E\exp \Big (\frac{|V_{\alpha ,f}|}{C_1\Vert f\Vert _p} \Big )^{\frac{N}{N-\alpha }}dx\le C_2\mu (E). \end{aligned}$$

(22.1c)

23c Some Consequences of Steiner’s Symmetrization

1.1 23.1c Applications of the Isodiametric Inequality

The Hausdorff outer measure $\mathcal {H}_{\alpha }$, introduced in § 5 of Chap. 3, can be properly re-normalized by a factor $\gamma _{\alpha }$, so that when $\alpha =N$ it coincides with the Lebesgue outer measure $\mu _e$ in $\mathbb {R}^N$. Set

$$\begin{aligned} \gamma _{\alpha }=\frac{{\displaystyle \pi ^{\frac{\alpha }{2}}}}{2^\alpha \varGamma \big (\frac{\alpha }{2}+1\big )} \qquad \text { where }\qquad \varGamma (t)= \int _0^\infty e^{-x} x^{t-1}dx,\quad t>0, \end{aligned}$$

is the Euler gamma function. One verifies that for $\alpha =N$

$$\begin{aligned} \gamma _N=\frac{\kappa _N}{2^N},\qquad \kappa _N=\{\text {volume of the unit ball in }\mathbb {R}^N\}. \end{aligned}$$

Define the re-normalized Hausdorff outer measure as Measure(s)!outer!Hausdorff!re-normalized

$$\begin{aligned} H_\alpha =\gamma _{\alpha } \mathcal {H}_{\alpha }. \end{aligned}$$

Proposition 23.1c

$H_N(E)=\mu _e(E)$ for all subsets $E\subset \mathbb {R}^N$.

Proof

We may assume that $\mu _e(E)<\infty$. Having fixed $\varepsilon >0$, let $\{E_j\}$ be a countable collection of sets in $\mathbb {R}^N$ each of diameter not exceeding $\varepsilon$ and such that $E\subset \cup E_j$. By the isodiametric inequality

$$\begin{aligned} \mu _e(E)\le \mathop {\textstyle {\sum }}\limits \mu _e(E_j)\le \mathop {\textstyle {\sum }}\limits \kappa _N({\textstyle \frac{1}{2}}{\text {diam }}{E_j})^N= \gamma _N\mathop {\textstyle {\sum }}\limits ({\text {diam }}{E_j})^N. \end{aligned}$$

Taking the infimum of all such collections $\{E_j\}$, and then letting $\varepsilon \rightarrow 0$, gives

$$\begin{aligned} \mu _e(E)\le \gamma _N\mathcal {H}_{\alpha }(E)= H_{\alpha }(E). \end{aligned}$$

For $\varepsilon >0$ fixed, let $\{Q_j\}$ be a countable collection of cubes in $\mathbb {R}^N$ with faces parallel to the coordinate planes, such that $E\subset \mathop {\textstyle {\bigcup }}\limits Q_j$ and

$$\begin{aligned} \mu _e(E)\ge \mathop {\textstyle {\sum }}\limits \mu _e(Q_j)-\varepsilon . \end{aligned}$$

By the Besicovitch measure theoretical covering of open sets in $\mathbb {R}^N$ (Proposition 18.1c of Chap. 3) for each $Q_j$ there exists a countable collection of disjoint, closed balls $\{B_{i,j}\}$, contained in $\buildrel {\mathrm{{o}}}\over {Q}_j$, of diameter not exceeding $\varepsilon$ such that

$$\begin{aligned} \mu _e\big (Q_j-\mathop {\textstyle {\bigcup }}\limits B_{i,j}\big )= \mu \big (Q_j-\mathop {\textstyle {\bigcup }}\limits B_{i,j}\big ) =\mu \big (\buildrel {\mathrm{{o}}}\over {Q}_j-\mathop {\textstyle {\bigcup }}\limits B_{i,j}\big )=0. \end{aligned}$$

For these residual sets, by Proposition 5.2 of Chap. 3,

$$\begin{aligned} \mathcal {H}_N\big (Q_j-\mathop {\textstyle {\bigcup }}\limits B_{i,j}\big )=0. \end{aligned}$$

Then compute and estimate

$$\begin{aligned} \begin{aligned} H_N(E)&\le \mathop {\textstyle {\sum }}\limits H_N(Q_j)\le {\mathop {\textstyle {\sum }}\limits }_j H_N\big ({\mathop {\textstyle {\bigcup }}\limits }_i B_{i,j}\big )\\&\le {\mathop {\textstyle {\sum }}\limits }_{i,j}\gamma _N({\text {diam }}{B_{i,j}})^N={\mathop {\textstyle {\sum }}\limits }_{i,j}\mu _e(B_{i,j})\\&=\mathop {\textstyle {\sum }}\limits \mu _e(Q_j)\le \mu _e(E)+\varepsilon . \end{aligned} \end{aligned}$$

$\blacksquare$